Second Degree Homogeneity

We now show that the second variation is homogeneous of the second degree in h, i. e., 

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From this become the second degree homogenous dissipation potentials F=i[A(T)v( ij].v( i], 4. = i[R(T)j ]-j Consequently the partially restricted variation-task of (73) in this actual case is... 

As the net result of multiplying each independent variable by the parameter X merely has been to multiply the function by X, the function is called homogeneous. Because the exponent of X in the result is 2, the function is of the second degree. 

A cone is a surface generated by straight lines which always pass through a fixed point (the vertex of the cone) and obey some other law. If P(x, y, z) be any point on the surface, then any other point (A x, ky, kz) on the line OP is also on the surface, so that if the vertex is taken as the origin, the equation of a cone is homogeneous, and hence the equation of a cone of the second degree (quadric cone) is ... 

This way we have established that the surface is described by a second-degree equation with constant coefficients in each homogeneous coordinate system. 

Method of Variation of Parameters This method is apphcable to any linear equation. The technique is developed for a second-order equation but immediately extends to higher order. Let the equation be y" + a x)y + h x)y = R x) and let the solution of the homogeneous equation, found by some method, he y = c f x) + Cofoix). It is now assumed that a particular integral of the differential equation is of the form P x) = uf + vfo where u, v are functions of x to be determined by two equations. One equation results from the requirement that uf + vfo satisfy the differential equation, and the other is a degree of freedom open to the analyst. The best choice proves to be... 

Every peak was uniform with respect to m, but each one had a distribution in block length with respect to the PEO blocks (n). To identify fractions, they were collected and subjected to mass spectrometry. The first fraction contained polyethylene glycol and block oligomers with a degree of polymerization m(PO) 1-3. The second fraction was homogeneous with respect to PO and contained m(PO) 4, while fraction 3 resulted from m(PO) = 5. 

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